Using functions with parameters

[Herman Rubin]

Suppose one has a procedure P which takes a function or functions
and produces a number or numbers. The numbers produced are NOT
integrals. Now suppose one has a function with a parameter (I
know that Maple does not have this, but does it as a function
of two arguments) and I want to produce a graph of the values
of P on the function as a function of the parameter. How to do
this without rewriting everything?


--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558

[Axel Vogt]

On 12.04.2013 00:12, Herman Rubin wrote:
> Suppose one has a procedure P which takes a function or functions
> and produces a number or numbers. The numbers produced are NOT
> integrals. Now suppose one has a function with a parameter (I
> know that Maple does not have this, but does it as a function
> of two arguments) and I want to produce a graph of the values
> of P on the function as a function of the parameter. How to do
> this without rewriting everything?

Do you mean the function F(p,.) = a |---> F(p,a) ?

F:= proc(x,a) sin(a*x) end proc; # example
plot( F(1, a), a= 0 .. 4*Pi);

[Herman Rubin]

It is not that simple. The procedure uses the function
at a finite number of arguments, computed recursively and
depending on the functions, to get two numbers.

[Axel Vogt]

On 12.04.2013 19:29, Herman Rubin wrote:
> On 2013-04-12, Axel Vogt <&nor...@axelvogt.de> wrote:
>> On 12.04.2013 00:12, Herman Rubin wrote:
>>> Suppose one has a procedure P which takes a function or functions
>>> and produces a number or numbers. The numbers produced are NOT
>>> integrals. Now suppose one has a function with a parameter (I
>>> know that Maple does not have this, but does it as a function
>>> of two arguments) and I want to produce a graph of the values
>>> of P on the function as a function of the parameter. How to do
>>> this without rewriting everything?
>
>> Do you mean the function F(p,.) = a |---> F(p,a) ?
>
>> F:= proc(x,a) sin(a*x) end proc; # example
>> plot( F(1, a), a= 0 .. 4*Pi);
>
>
> It is not that simple. The procedure uses the function
> at a finite number of arguments, computed recursively and
> depending on the functions, to get two numbers.

Ok, but I can not imagine what is meant exactly. Would you mind
to provide a formal example (i.e. a procedure, which works like
your actual problem) ?

[Herman Rubin]

Here is the procedure which I constructed; it works.
In practice, n will be 128 or 256.

eff:=proc(f,n,start,F)
local s1,s2,i,x,h;
s2:=0;x:=start;
for i from 1 to n do
h:=f(x);
x:= x+ 1/(n*f(x));
s2:=s2+(1-f(x)/h);
end do;
s2:=s2/n;
s1:=F(x);
return s1,s2;
end proc;

[Axel Vogt]

On 12.04.2013 22:22, Herman Rubin wrote:
> On 2013-04-12, Axel Vogt <&nor...@axelvogt.de> wrote:
>> On 12.04.2013 19:29, Herman Rubin wrote:
>>> On 2013-04-12, Axel Vogt <&nor...@axelvogt.de> wrote:
>>>> On 12.04.2013 00:12, Herman Rubin wrote:
>>>>> Suppose one has a procedure P which takes a function or functions
>>>>> and produces a number or numbers. The numbers produced are NOT
>>>>> integrals. Now suppose one has a function with a parameter (I
>>>>> know that Maple does not have this, but does it as a function
>>>>> of two arguments) and I want to produce a graph of the values
>>>>> of P on the function as a function of the parameter. How to do
>>>>> this without rewriting everything?

...

> Here is the procedure which I constructed; it works.
> In practice, n will be 128 or 256.
>
> eff:=proc(f,n,start,F)
> local s1,s2,i,x,h;
> s2:=0;x:=start;
> for i from 1 to n do
> h:=f(x);
> x:= x+ 1/(n*f(x));
> s2:=s2+(1-f(x)/h);
> end do;
> s2:=s2/n;
> s1:=F(x);
> return s1,s2;
> end proc;

I would amend it to return a pair (and personally only return s2, since s1 = F(s2)).
Likewise I would use a subsequent proc to do modify the output.

An 'option hfloat' will speed up computations (the code can be improved a bit more).

Eff:=proc(f,n,start,F)
option hfloat; # to speed up numerics

local s1,s2,i,x,h;
s2:=0;x:=start;
for i from 1 to n do
h:=f(x);
x:= x+ 1/(n*f(x));
s2:=s2+(1-f(x)/h);
end do;
s2:=s2/n;
s1:=F(x);

return [s1,s2]; # return as pair
end proc;


# a small test
Eff(exp, 8, 2.1, 'x -> x^2');

[4.91205739100430, -0.0146458557860588]

Then I can plot (using only few data points by option 'numpoints')

nTst:=8; fTst:=ln; FTst:='x -> x';
plot( Eff(fTst, nTst, s, FTst), s=1 ..4, numpoints=5);

[Herman Rubin]

The pair idea may or may not be useful. However, s1 is NOT F(s2); s1
will always be less than s2, and in most cases, I believe about half.

> Likewise I would use a subsequent proc to do modify the output.



> An 'option hfloat' will speed up computations (the code can be improved a bit more).

> Eff:=proc(f,n,start,F)
> option hfloat; # to speed up numerics
> local s1,s2,i,x,h;
> s2:=0;x:=start;
> for i from 1 to n do
> h:=f(x);
> x:= x+ 1/(n*f(x));
> s2:=s2+(1-f(x)/h);
> end do;
> s2:=s2/n;
> s1:=F(x);
> return [s1,s2]; # return as pair
> end proc;


> # a small test
> Eff(exp, 8, 2.1, 'x -> x^2');

> [4.91205739100430, -0.0146458557860588]

A test, but not appropriate It was designed for f a decreasing
probability density, and F the corresponding tail probability.

> Then I can plot (using only few data points by option 'numpoints')

> nTst:=8; fTst:=ln; FTst:='x -> x';
> plot( Eff(fTst, nTst, s, FTst), s=1 ..4, numpoints=5);

This is a misconception of what is wanted. What is wanted
is to have a parametric family of densities, and plot the
values as a function of the parameters. This means that
eff has to be entered with a different function for each
value of the parameter.